Poisson-Lie interpretation of trigonometric Ruijsenaars duality

Abstract

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of `free' Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars' proof of the crucial symplectomorphism property of the duality map.

modified Theorem 3.1 and added new results in v2, 49 pages; v3: final version (with a reference added) to appear in CMP

References in corpus (4)

• On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
• A class of Calogero type reductions of free motion on a simple Lie group
• Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction
• On moment maps associated to a twisted Heisenberg double

Cited by in corpus (29)

• On Three Dimensional Quiver Gauge Theories and Integrability
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• The hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality
• BPS States in Omega Background and Integrability
• Poisson-Lie analogues of spin Sutherland models
• Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities
• New compact forms of the trigonometric Ruijsenaars-Schneider system
• Action-angle duality between the C(n)-type hyperbolic Sutherland and the rational Ruijsenaars-Schneider-van Diejen models
• The action-angle dual of an integrable Hamiltonian system of Ruijsenaars--Schneider--van Diejen type
• Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system
• Scattering theory of the hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars--Schneider--van Diejen models
• On the origin of dual Lax pairs and their $r$-matrix structure
• Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters
• The full phase space of a model in the Calogero-Ruijsenaars family
• Quantum Trace Formulae for the Integrals of the Hyperbolic Ruijsenaars-Schneider model
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• On a Poisson-Lie deformation of the BC(n) Sutherland system
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
• Poisson reductions of master integrable systems on doubles of compact Lie groups
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• The Ruijsenaars self-duality map as a mapping class symplectomorphism
• On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system
• An integrable BC(n) Sutherland model with two types of particles
• An application of the reduction method to Sutherland type many-body systems
• Integrable many-body systems of Calogero-Ruijsenaars type
• Poisson-Lie analogues of spin Sutherland models revisited
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction